Goodness-of-fit tests for the error distribution in nonparametric regression

Suppose the random vector (X,Y) satisfies the regression model Y=m(X)+σ(X)ε, where m(⋅) is the conditional mean, σ²(⋅) is the conditional variance, and ε is independent of X. The covariate X is d-dimensional (d≥1), the response Y is one-dimensional, and m and σ are unknown but smooth functions. Goodness-of-fit tests for the parametric form of the error distribution are studied under this model, without assuming any parametric form for m or σ. The proposed tests are based on the difference between a nonparametric estimator of the error distribution and an estimator obtained under the null hypothesis of a parametric model. The large sample properties of the proposed test statistics are obtained, as well as those of the estimator of the parameter vector under the null hypothesis. Finally, the finite sample behavior of the proposed statistics, and the selection of the bandwidths for estimating m and σ are extensively studied via simulations.